On Rigid Matrices and U-Polynomials

We introduce a class of polynomials, which we call U-polynomials, and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of U-polynomials, though their size is larger than desired. Furthermore, we give two alternative proofs for the fact that small-bias sets induce rigid matrices.Finally, we construct rigid matrices from unbalanced expanders, with essentially the same size as the construction via small-bias sets.

[1]  Enkatesan G Uruswami.  Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes , 2008 .

[2]  Avi Wigderson,et al.  Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes , 2010, STOC '11.

[3]  Amnon Ta-Shma,et al.  Constructing Small-Bias Sets from Algebraic-Geometric Codes , 2009, FOCS.

[4]  Noga Alon,et al.  Random Cayley Graphs and Expanders , 1994, Random Struct. Algorithms.

[5]  Kousha Etessami,et al.  Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations , 2005, JACM.

[6]  Satyanarayana V. Lokam.  Spectral methods for matrix rigidity with applications to size-depth tradeoffs and communication complexity , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[7]  Noga Alon,et al.  Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs , 1992, IEEE Trans. Inf. Theory.

[8]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[9]  T. Sanders,et al.  Analysis of Boolean Functions , 2012, ArXiv.

[10]  GuruswamiVenkatesan,et al.  Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes , 2009 .

[11]  A. Razborov,et al.  Improved lower bounds on the rigidity of Hadamard matrices , 1998 .

[12]  Daniel A. Spielman,et al.  A Remark on Matrix Rigidity , 1997, Inf. Process. Lett..

[13]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[14]  Vikraman Arvind,et al.  The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets , 2010, STACS.

[15]  Vojtech Rödl,et al.  Pseudorandom sets and explicit constructions of ramsey graphs , 2004 .

[16]  Joel Friedman,et al.  A note on matrix rigidity , 1993, Comb..

[17]  Noga Alon,et al.  Simple construction of almost k-wise independent random variables , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[18]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[19]  Noga Alon,et al.  Deterministic Approximation Algorithms for the Nearest Codeword Problem , 2009, APPROX-RANDOM.

[20]  Ryan O'Donnell,et al.  Analysis of Boolean Functions , 2014, ArXiv.

[21]  Satyanarayana V. Lokam.  Complexity Lower Bounds using Linear Algebra , 2009, Found. Trends Theor. Comput. Sci..

[22]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[23]  Noga Alon,et al.  Explicit construction of linear sized tolerant networks , 1988, Discret. Math..

[24]  Zeev Dvir,et al.  On Matrix Rigidity and Locally Self-correctable Codes , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.

[25]  Amnon Ta-Shma,et al.  Constructing Small-Bias Sets from Algebraic-Geometric Codes , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[26]  Leslie G. Valiant,et al.  Graph-Theoretic Arguments in Low-Level Complexity , 1977, MFCS.